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**Pricing dynamic investment fund protection (With discussion by Terence Chan, François-Serge Lhabitant and Svein-Arne Persson and a reply by the authors).**
*(English)*
Zbl 1083.91516

Summary: We consider an investment fund whose unit value is modeled by a geometric Brownian motion. Different forms of dynamic investment fund protection are examined. The basic form is a guarantee that instantaneously provides the necessary payments so that the upgraded fund unit value does not fall below a protected level. A closed form expression for the price of such a guarantee is derived. This result can also be applied to price a guarantee where the protected level is an exponential function of time. Moreover, it is explicitly shown how the protection can be generated by construction of the replicating portfolio. The dynamic investment fund protection is compared with the corresponding put option, and it is shown that for short time intervals the ratio of the prices approaches 2. Finally, a more exotic guarantee is considered, where the protected level is a given percentage of the maximal observed fund unit value. Assuming that the protected level remains constant once the payments have started, we obtain a surprisingly simple formula for the price of a perpetual guarantee. Several numerical and graphical illustrations show how the theoretical results can be implemented in practice.

### MSC:

91B28 | Finance etc. (MSC2000) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

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\textit{H. U. Gerber} and \textit{G. Pafumi}, N. Am. Actuar. J. 4, No. 2, 28--41 (2000; Zbl 1083.91516)

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### References:

[1] | Abramowitz M., Handbook of Mathematical Functions (1972) · Zbl 0543.33001 |

[2] | Baxter M., Financial Calculus (1996) |

[3] | Cox D.R., The Theory of Stochastic Processes (1965) · Zbl 0149.12902 |

[4] | Dothan M., Prices in Financial Markets (1990) · Zbl 0744.90010 |

[5] | Gerber H.U., North American Actuarial Journal 2 (3) pp 101– (1998) · Zbl 1081.91528 |

[6] | Gerber H.U., Insurance: Mathematics and Economics 24 (1) pp 3– (1999) · Zbl 0939.91065 |

[7] | Panjer H., Financial Economics: With Applications to Investments, Insurance and Pensions (1998) |

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